At some point in their life everyone needs to calculate the vertical component of four force vectors acting on a rigid body, given the horizontal components of those vectors. In school I learned how to solve this problem with two forces, in two dimensions, and that method can be easily extended to three forces in three dimensions. But, the method produces a system of three linear equations, with four unknowns…that doesn’t work.
At this point I could look up techniques, but I’m a red neck with a love for the numbers, so I’m gonna figure it out on my own. My method started as simply saying that each force is proportional to some infinitesimal deflection where the force “touches” the body, in addition to the translation and rotation assumptions normally applied. The first steps toward this solution were great, I just applied a linear transform to translate and rotate each point in the body: . From there its just a quick hop down derivative alley, a turn at and bam! infinitesimal deflection:
Sure, it’d be great if it stopped there, but no. Since the deflection problem translated a dot product into a cross product, I thought maybe the torque (cross product) and moment of inertia equations could be derived from work (dot product). They very likely can, but after a day and a half of recalculating around transposition errors, self deprecation is starting to weigh on me. In spite of the voices saying ‘I should have just looked it up’, I’m gonna stay the course…it’s tradition.
At any rate, Jan’s getting close and has this to show for it.